### Polytropic Process(1)

```THERMODYNAMICS LAB
Mass and Energy Analysis of
Control Volumes
Polytropic Processes
ENTC - 370
ENTC 370
1
Polytropic Process(1)
• During expansion and compression processes
of gases, the following relationship holds:
PV n = C1
If m is constant, m n is also constant
Vn
P n = C2
m
Pv n = C2
Equation_ 1
For a Process from state 1 to state 2:
P1v1n = P2 v2n
• The coefficient n depends on the process.
2
Polytropic Process(2)
• During expansion and compression processes
of gases, the following relationship holds:
Taking
natural log_ of_ Equation_
1:
ln(P)  n ln( v )  C
Y  ln(P);
X  ln( v )
Y  -n X  C
If we measure
T and P, we can_ obtain_
PV  mRT 
V
m

RT
P
 v
v_ from_ Ideal_ gas_ equation
RT
P
• The coefficient n depends on the process.
3
:
Polytropic Process
• The coefficient n depends on the process:
― n=0 , Isobaric process (constant pressure) 5-1 in graph.
― n=∞, Isometric process (constant volume) 2-6 in graph.
― n=1, Isothermal process (constant temperature) 4-8 in
graph.
― n=k, Adiabatic process (no heat transfer) 3-7 in the
graph. k=cp/cv=1.4 for air.
Graph from www.taftan.com
ENTC 370
4
Polytropic Process
• Boundary work:
Wb 
m R (T 2  T1 )
1 n
 V2 
W b  P V ln 

 V1 
ENTC 370
n 1
n= 1
5
Problem 1: Polytropic Process (Excel)
-
Pressurized air inside a pressure vessel is expanded in a polytropic
process using three discharge valves with small, medium and large
orifices. The measured temperature and pressure for the process
are posted.
1. Use the ideal gas law, Pv = RT, to compute v for each
corresponding P. Use SI units: m3/kg for v, kPa for P and ºK for T.
Conversion factor: 6.894 kPa=1 PSI
ºK = ºC+273.15
R= 0.286 KJ/(kg ºK) for air
2. Plot ln(P) versus ln(v) and find n:
a. For each run, on a separate graph, plot ln(P) [on the ordinate
(vertical) axis] versus ln(v) [on the abscissa (horizontal) axis].
b. Determine the polytropic exponent n by using a linear model of
each run. Also find the correlation coefficient R2.
3. Discuss the meaning of your n values, that is, how do the n values
compare with n values for other, known processes (see previous
slide)?
ENTC 370
6
Turbines and Compressors
• Analysis for steady state systems, Energy
balance:
Turbines
:
E in  E out
v1
2
2
v2

m ( h1 
 gz 1 )  W out  m ( h 2 
 gz 2 )
2
2
 ke 
v v
2
1
2
2
2
ENTC 370
7
Problem 2: Steam Turbine (EES)
Steam flows steadily (8 kg/sec, mass flow
rate) through an adiabatic turbine. The inlet
conditions of the steam are 10 MPa, 350 ºC,
and 65 m/sec. The exit conditions are 85%
quality, and 40 m/sec. The exit pressure
varies from 10 kPa to 200 kPa.
P1,T1,V1
Determine:
-Change in Kinetic Energy (ke)
-Turbine inlet area
P2,x2,V2
-Plot the power output against the outlet
pressure
ENTC 370
8
Problem 2: Steam Turbine (EES)
m 
V


vA

where ,
m  mass _ flow _ rate
V  Volume _ flow _ rate
v  velocity
  specific _ volume
A  cross _ sec tional _ area
ENTC 370
9
Individual Lab Report
• Introduction: Briefly explain the objectives of the